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JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 45
Estimating Subsurface Velocities from Surface Fields with Idealized
Stratification
J. H. LACASCE
Department of Geosciences, University of Oslo, Oslo, Norway
J. WANG
Scripps Institute of Oceanography, University of California, San Diego, La Jolla, California
(Manuscript received 16 October 2014, in final form 20 June 2015)
ABSTRACT
A previously published method by Wang et al. for predicting subsurface velocities and density from sea
surface buoyancy and surface height is extended by incorporating analytical solutions to make the vertical
projection. One solution employs exponential stratification and the second has a weakly stratified surface
layer, approximating a mixed layer. The results are evaluated using fields from a numerical simulation of the
North Atlantic. The simple exponential solution yields realistic subsurface density and vorticity fields to
nearly 1000 m in depth. Including a mixed layer improves the response in the mixed layer itself and at high
latitudes where the mixed layer is deeper. It is in the mixed layer that the surface quasigeostrophic approximation is most applicable. Below that the first baroclinic mode dominates, and that mode is well approximated by the analytical solution with exponential stratification.
1. Introduction
Wang et al. (2013, hereinafter W13) proposed a
method for projecting surface density and height
downward in the water column. The method requires
simultaneous observations of surface density (or temperature, in the absence of salinity) and height. The
density projection is made using the surface quasigeostrophic (SQG) approximation (Blumen 1978; Held
et al. 1995; Lapeyre and Klein 2006; LaCasce and
Mahadevan 2006; Tulloch and Smith 2006; IsernFontanet et al. 2008). The height is then used to deduce
the two gravest baroclinic modes. W13 found that the
SQG portion was most important in the near-surface
region, while the baroclinic modes dominated at depth.
Here we simplify the method by using analytic solutions
for the vertical projection. This obviates determining the
Corresponding author address: Joe LaCasce, Department of
Geosciences, University of Oslo, P.O. Box 1022 Blindern, N-0315
Oslo, Norway.
E-mail: j.h.lacasce@geo.uio.no
DOI: 10.1175/JPO-D-14-0206.1
Ó 2015 American Meteorological Society
SQG and baroclinic mode solutions numerically. We also
consider a solution with a surface ‘‘mixed layer,’’ allowing us to gauge the latter’s effect on the construction.
2. Method
The method employs a two-component decomposition
of the quasigeostrophic potential vorticity (Pedlosky 1987):
!
› f02 ›
c 5 Q,
= c1
›z N 2 ›z
2
(1)
where N(z) is the Brunt–Väisälä frequency, c 5 p/(f0r0)
is the geostrophic streamfunction, and Q is the potential
vorticity (PV). Because the PV equation is linear, the
solution can be written as a superposition. The homogeneous solution is the SQG streamfunction cs, while
the particular solution is the ‘‘interior’’ streamfunction
ci (Lapeyre and Klein 2006). These have different surface boundary conditions:
›
c 5 0,
›z i
b
›
cs 5 s ,
›z
f0
at z 5 0,
(2)
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LACASCE AND WANG
where bs is the surface buoyancy. Thus, only the SQG
solution is directly linked to the surface density.
For the bottom boundary condition, W13 demanded
that the vertical derivative of both streamfunctions
vanish and that the total velocity be zero at the bottom. Alternately, one can simply require that each
streamfunction vanish with depth (e.g., Lapeyre and
Klein 2006; LaCasce and Mahadevan 2006; Lapeyre
2009):
^ 5c
^ 5 0.
lim c
i
s
z /2‘
(7)
^ exp(ikx 1 ily) and k 5 (k2 1 l2 )1/2 is the
Here c 5 åk,l c
total wavenumber. The PV and surface buoyancy are
^ s can be
similarly transformed. The SQG solution c
shown to be
^ 5
c
s
(3)
It turns out that using this condition greatly simplifies
the subsequent solutions.
The interior solution cannot be determined because
the interior PV Q is unknown, but the two gravest baroclinic modes can be deduced if the surface pressure is also
known (W13). The baroclinic modes are solutions of the
Sturm–Liouville problem:
!
› f02 ›
F 1 R22
n Fn 5 0,
›z N 2 ›z n
^ 2 dc
^ N 2 k2
N02 k2 ^ 2z/h
d2 c
2z/h ^
0
2
Qe
2
e
.
c
5
dz2 h dz
f02
f02
b^s z/h I1 (Le kez/h )
e
,
I0 (Le k)
N0 k
(8)
where the In are modified Bessel functions and Le [
N0h/f0 is a deformation-like scale associated with the
exponential stratification. The baroclinic modes, on the
other hand, satisfy
d2 Fn 2 dFn
N2
1 2 0 2 e2z/h Fn 5 0.
2
2
h dz Rn f0
dz
(9)
The solution that decays with depth has the form
(4)
Fn } e
z /h
J1
Le z/h
,
e
Rn
(10)
given
ci (x, y, z, t) 5
å gn (x, y, t)Fn (z) ,
(5)
where J1 is a Bessel function of the first kind. Imposing
the surface condition [(2)] yields
n
with the same boundary conditions as for ci. Here Rn is
the nth deformation radius and the g n are the modal
coefficients.
With zero flow at depth, the baroclinic modes are surface intensified and the barotropic mode is absent
(Pedlosky 1987; Samelson 1992; LaCasce 2012).1 As such,
there is only a single unknown, the amplitude of the first
baroclinic mode g 1, which can be determined from the
surface elevation h:
cs (x, y, 0, t) 1 ci (x, y, 0, t) ’ cs (x, y, 0, t) 1 g1 F1 (0)
g
5 h(x, y, t) .
f0
(6)
a. Exponential stratification
The SQG solution and baroclinic modes can be derived analytically for certain idealized stratifications.
One is an exponential: N 5 N0ez/h. Using this in (1)
yields
J0
Le
5 0.
Rn
(11)
So, the Rn are determined from the zeros of J0. The first
is 2.4048, so
R1 5
Le
.
2:4048
(12)
Notice that the eigenvalue problem is solved without a
transcendental equation. This is because of the choice of
lower boundary condition.
The full solution is then
^5
c
b^s z/h I1 (Le kez/h )
e
1 g 1 ez/h J1 (2:4048ez/h ) .
I0 (Le k)
N0 k
(13)
We determine g1 from (6):
"
#
b^s I1 (Le k)
1
g^
h
.
g1 5
2
J1 (2:4048) f0
N0 k I0 (Le k)
(14)
b. Mixed layer
1
Over steep topography the barotropic mode is replaced by a
bottom-intensified topographic wave mode, which generally cannot be deduced from surface information alone.
We can make the stratification more realistic by
adding a surface mixed layer:
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JOURNAL OF PHYSICAL OCEANOGRAPHY
(
N5
Nm
ND e
(z1D)/h
if
z . 2D
if
z# 2D
VOLUME 45
.
The mixed layer lies above z 5 2D, and generally
Nm ND. The stratification N is thus discontinuous at
z 5 2D.
The solution in this case is more involved and is left for
the appendix. It consists of a constant stratification solution in the mixed layer and an exponential solution
below z 5 2D, which are then matched at z 5 2D. The
streamfunction is matched, so that the horizontal velocities are continuous. We also match (›c/›z)/N 2 ,
which guarantees continuity of the vertical velocity. This
follows from the quasigeostrophic density equation:
f d ›c
.
w 5 2 02
N dt ›z
(15)
Doing this also makes the PV continuous at z 5 2D.
It will be seen, however, that this condition yields
unrealistic solutions, particularly when the mixed layer
stratification is weak, so we tested matching the buoyancy instead at the mixed layer base (as would be done
in the absence of a discontinuity in N). This produced
more realistic density variations both near and below the
mixed layer base.
3. Results
We evaluate the analytical solutions using fields from
the same North Atlantic simulation discussed by W13.
Full model details are given therein. The three regions
lie in the western and eastern Atlantic, and in the subpolar gyre (Fig. 1). In each region we average the density
laterally to obtain a profile for N(z) and use the result to
fit the analytical N curves. We also calculate the rms
density and vorticity as functions of depth, for comparison with the solutions.
The three stratification profiles, with the two idealized
fits, are plotted in the left column of Fig. 2. The exponentials were obtained by fitting the deeper portion of
N, below the region of rapid variation in the upper
several hundred meters. For the mixed layer solution,
the stratification Nm was obtained for the shallowest,
weakly stratified layer, while the depth D was determined from the position of the maximum of N (about
50 m in regions 1 and 2 and 400 m in region 3). All parameters are listed in Table 1. The values of the deformation radius R1 range from 22.6 km (region 3) to
32 km (region 2).
The results in Fig. 2 correspond to the second mixed
layer solution in the appendix, in which ›c/›z is matched
at z 5 2D. This requires solving a transcendental
FIG. 1. Surface density from the North Atlantic in the Parallel
Ocean Program (POP) simulation of W13. The three regions to be
considered are indicated by the stippled boxes (from W13).
equation, given in (A12). However, the first root varies
little for reasonable values of the mixed layer stratification and depth, as indicated in Table 1. Thus, R1 is well
approximated by the pure exponential result [(12)], with
the deep stratification ND replacing the surface value N0.
As such, neither analytical profile requires a numerical
solution for the baroclinic mode.
Shown in the other columns of Fig. 2 are the standard
deviations of the density and vorticity plotted against
depth. The results for the exponential and mixed layer
solutions are shown, as are the curves obtained by W13.
The latter derive from a numerical solution for the
baroclinic modes, using the actual stratification shown in
the left panels.
In region 1, the vorticity deviations (Fig. 2c) are of
similar magnitude for all three solutions, down to
roughly 700 m. The density deviations (Fig. 2b) are also
similar, and the solutions capture the subsurface maximum seen near 250-m depth. The exponential and
mixed layer solutions behave much the same, though
the latter is better in the mixed layer itself; below that,
the two yield very similar results. Moreover, both analytical solutions are as successful as the numerical
solution of W13.
Similar comments apply in region 2 (Figs. 2d–f). The
vorticity variations are somewhat better for the simple
exponential solution, although the differences from the
other solutions are small. Interestingly, the two analytical solutions perform better than the full numerical
solution in terms of density, as the latter yields much
greater variations with depth. Evidently, the analytical
fit smooths out the small-scale structures in N, which
have little impact on the density variations. Again, the
mixed layer solution is most successful in the mixed
layer itself.
SEPTEMBER 2015
LACASCE AND WANG
FIG. 2. The horizontally averaged fields from the three regions shown in Fig. 1: (a)–(c) region
1, (d)–(f) region 2, and (g)–(i) region 3. Shown are (left) N2, (middle) the rms density anomaly,
and (right) the rms vorticity anomaly. The exponential and mixed layer solutions are shown in
black and red contours, respectively, and the results from W13 are shown in green. The model
fields are indicated by blue plus symbols.
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VOLUME 45
TABLE 1. Parameters for the two N2 fits. The depth range for the fitting is [380, bottom]. EXP and WML represent the cases with a pure
exponential and with a mixed layer overlying an exponential, respectively. The deformation radii come from (12) (EXP) and (A12)
(WML). The second estimate for WML (in parentheses) stems from using the exponential estimate [(12)], with Nm replacing N0.
1
2
3
Case
EXP
WML
EXP
WML
EXP
WML
N0 (s21)
ND (s21)
Nm (s21)
h (m)
MLD (m)
f0
R1 (km)
0.0072
—
—
770
—
9.68 3 1025
23.8
—
0.0068
0.0019
770
50
9.68 3 1025
20.7 (20.8)
0.0085
—
—
690
—
7.62 3 1025
32.0
—
0.0078
0.001
690
60
7.62 3 1025
26.7 (26.7)
0.0031
—
—
2000
—
1.14 3 1024
22.6
—
0.0026
0.0005
2000
400
1.14 3 1024
15.1 (15.3)
Region 3 differs because the mixed layer is substantially
deeper, extending to roughly 400 m. The mixed layer solution accordingly performs better. The predicted density
variations are nearly depth independent in the mixed
layer itself, as in the model. The exponential solution instead yields density variations that decay monotonically
with depth. The full numerical solution varies even more
with depth, but in this case the variations are realistic. It
performs best below 1000 m, where the analytical solutions are less accurate. However, the differences for the
vorticity deviations are much less. All three solutions yield
reasonable estimates in the upper 500 m.
Thus, the analytical solutions are generally as good as
the reconstructions of W13, despite their idealizations.
Including the surface mixed layer improves the density
variations in the mixed layer itself but has relatively little
impact on the horizontal velocities. Below the mixed
layer the simpler exponential fit yields equally good fits
for both density and vorticity.
As noted, the mixed layer solution uses an improper
matching condition on ›c/›z at the base of the mixed
layer. The result is that the density is continuous at
z 5 2D but the vertical velocity is not. Matching w instead yields the solution in (A6). The two mixed layer
solutions are compared in Fig. 3, using the region 3 fields.
Matching (›c/›z)/N2 instead of ›c/›z produces a
discontinuity in the density variations at the mixed
layer base. The variations in the mixed layer in the
former solution are smaller than in the latter, and they
are much larger below. The vorticity deviations in the
mixed layer are also better captured by the solution
with a continuous density; the other solution grossly
overestimates the variations. But while both solutions
produce too large deviations below the mixed layer, the
continuous w solutions are much greater, producing the
appearance of a discontinuity at the mixed layer base.
The curve is actually continuous, but the deviations
increase greatly over a small depth range.
FIG. 3. Comparing the two mixed layer solutions, the solution in which ›c/›z is matched at the
mixed layer base (in green) and that in which (›c/›z)/N2 is matched instead (in purple). The fields
are from region 3, and the panels shown (left) N2, (middle) rms density, and (right) rms vorticity.
SEPTEMBER 2015
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2429
FIG. 4. Plan views from region 1 of the density anomaly at 520-m depth for (a) the model, (b) the pure
exponential fit, and (c) the exponential with the mixed layer. (d)–(f) Vertical slices of the density anomaly
along the dashed line indicated in the upper panels.
A third solution was also tested, in which N was assumed to increase exponentially to the base of the mixed
layer and decay exponentially below (see the appendix).
While the results (not shown) were better than with the
continuous w solution above, they were still significantly
worse than with the continuous density solution.
Thus, we retain the mixed layer solution that matches
›c/›z at z 5 2D. This is obviously a practical choice
rather than a rigorous one, as the solution implies discontinuous vertical velocities. The unrealistic element is
the discontinuity in N, as the model profiles are instead
continuous. We retain the continuous density solution in
the interest of having a relatively simple analytical profile.
Further comparisons are shown in the subsequent figures. Snapshots of the density anomaly in region 1 are
shown in Fig. 4. The model fields are in the left column
and the exponential and mixed layer solutions are in the
middle and right columns, respectively. Figures 4a–c show
plan views of the density anomalies at 520-m depth, and
Figs. 4d–f show density cross sections taken along the
dashed line in the upper panels, near 40.58N.
Both analytical solutions capture the horizontal structure and amplitude of the model anomalies (Figs. 4a–c),
beyond relatively minor regional differences. Note that
520 m is well below the mixed layer depth (roughly 50 m).
The vertical structure (Figs. 4d–f) is also very similar. The
analytical solutions decay more slowly with depth, a
consequence of using only a single baroclinic mode in the
decomposition. Nevertheless, the overall picture is very
similar. The mixed layer solution is slightly better in the
mixed layer itself, capturing, for example, the vertical
variations near 438W, but the exponential solution is
basically as good.
The comparisons in region 2 are very similar and are not
shown. The fields for region 3 are contoured in Fig. 5. The
depth (520 m) for the plan views is just below the mixed
layer (400 m). Again, the structures and amplitudes are
very similar, outside of regional variations. Differences are
more apparent in the vertical though, with the mixed layer
solution clearly superior at capturing the variations in and
just below the mixed layer. Again, the decay below the
mixed layer is too gradual in both solutions because of
having only a single baroclinic mode.
The vorticity fields compare similarly. In region 1
(Fig. 6), the two solutions produce realistic structures
with reasonable amplitudes. The vertical slices too are
very similar, above 800 m. However, the model anomalies extend deeper than in the solutions, which is unsurprising given that the barotropic component is absent
in the solutions. But the result is good in the upper part
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VOLUME 45
FIG. 5. As in Fig. 4, but for region 3.
of the water column, and there is no appreciable improvement with the mixed layer model.
Similar comments apply in region 3 (Fig. 7). The solutions are comparable in the upper 1000 m, but the
model vorticity anomalies extend deeper. And here too,
the mixed layer yields only minor changes from the pure
exponential.
The similarities between the solutions and the model
are quantified in Fig. 8, which shows the correlations
between the analytical solutions and the model density
(left) and vorticity (right) as functions of depth. In region 1 (top panels), the correlations for the density for
both solutions (solid contours) are between 0.8 and 1.0
in the upper 1000 m. The correlations for the vorticity
are similarly high. Moreover, the two solutions are not
greatly different.
W13 found that the SQG contribution to the full solution was generally less than that of the baroclinic
modes. We examined this by calculating the correlations
for the SQG portions alone (dashed contours in Fig. 8).
The correlations for the density are comparable to those
for the full solutions in the upper 50 m. This is as expected, since the SQG solution matches the model
density at the surface, but the correlations decrease
below the mixed layer; at 500 m they are nearer to 0.5 for
both analytical profiles. Of course, the correlations do
not reflect the amplitude of the SQG contribution, and
the latter decrease even more rapidly in comparison to
the model’s (not shown). The correlations for the SQG
vorticity (right panel) are lower even at the surface,
being slightly less than 0.6. This reflects that the surface
density is not always aligned with the surface pressure
(W13). Thus, the baroclinic mode contribution is more
important in this regard.
The results are qualitatively the same in the other
two regions (middle and bottom panels). It is striking
in particular that the two idealized profiles yield such
similar correlations. Despite that the mixed layer improves the amplitude of the density variations in the
mixed layer, it changes the structures little.
4. Summary and conclusions
We extended the study of W13 for predicting subsurface velocities and density from surface buoyancy
and sea surface height. We use analytical solutions to
make the vertical projection, assuming either an exponential N profile or an exponential with a mixed layer at
the surface. The solution is a combination of an SQG
component and the first baroclinic mode.
The solutions perform remarkably well in comparison
with subsurface fields from a North Atlantic simulation.
Both the density and vorticity fields are realistic, above
about 1000 m depth. Moreover, the comparisons are
SEPTEMBER 2015
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LACASCE AND WANG
FIG. 6. Plan views of the vorticity (normalized by f0) and vertical slices of the same quantity at 520-m depth in
region 1.
generally as good as those of W13, who used numerical
solutions to obtain the vertical dependence.
Several additional points are of interest. Including a
surface mixed layer improves the density fields in the
mixed layer itself but has little effect below. Also, both
solutions yield similar vorticity fields. This suggests
that a simple exponential profile may suffice in many
applications, and this can be easily determined from
climatological density.
Second, the reconstructions require only a single
baroclinic mode. We have dispensed with the barotropic
mode and accordingly miss variability at great depths,
but this is to be expected as the fields are reconstructed
solely from surface data. Nevertheless, if a barotropic
mode is desired, a flat bottom condition can be applied,
as with the exponential N solution of LaCasce (2012).
Then the two unknown modal amplitudes would be
determined as in W13. The price would be having to
solve a transcendental equation for the baroclinic
modes, something we have avoided here.
The mixed layer solution has a discontinuity at the
base of the mixed layer, and the solutions require
matching conditions there. Matching the density rather
than the vertical velocity yields better results, despite
that the latter is theoretically preferable. The reasoning
is that the weakly stratified mixed layer is always joined
to the stratified interior via a transition region, so that
the density is continuous. Including such a layer in the
theoretical model is possible but would defeat the purpose of having a simplified solution.
We found too that the SQG portion of the solution is
most important in the mixed layer, where the stratification is weak and the PV near zero. Thus, the SQG
construction is probably most valuable as an idealized
representation of mixed layer flow.
Acknowledgments. It is with pleasure and sadness that
we acknowledge Bach Lien Hua, a good friend and an
inspiration. She made major contributions to oceanography, in problems like the one considered here. Thanks
to two anonymous reviewers, Paola Cessi, Bill Young,
and Jörn Callies for diverse comments. The work was
supported in part by NSF Grant OCE-1234473 (Wang)
and by the Norwegian Research Council Grant 221780,
NORSEE (LaCasce).
APPENDIX
The Mixed Layer Solutions
The stratification is
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VOLUME 45
FIG. 7. As in Fig. 6, but for region 3.
(
N5
Nm
ND e
(z1D)/h
if
z . 2D
if
z# 2D
Nm
D 5 Nm k
cosh(Lm k)I0 (LD k) 1 sinh(Lm k)I1 (LD k) ,
ND
.
(A5)
The SQG solution in the mixed layer that satisfies the
upper boundary condition is
Nm kz
Nm kz
b^
^
sinh
1
.
c 5 A1 cosh
Nm k
f0
f0
(A1)
The solution in the lower region, which vanishes with
depth, follows from (8):
^ 5 A e(z1D)/h I [L ke(z1D)/h ],
c
2
1 D
(A2)
^
with LD [ NDh/f0. We obtain A1 and A2 by matching c
2
^
at z 5 2D. As noted, this guarantees
and (›c/›z)/N
continuity of the horizontal and vertical velocities, respectively. The result is
A1 5
b^ Nm
sinh(Lm k)I0 (LD k) 1 cosh(Lm k)I1 (LD k) ,
D ND
(A3)
b^
A2 5 , and
D
(A4)
with Lm [ NmD/f0. In the limit of vanishing surface
stratification, c is nearly barotropic in the mixed layer
and proportional to the surface buoyancy.
The derivation for the baroclinic modes is similar. The
upper and lower solutions that satisfy the boundary
conditions are
8
Lm z
>
>
B cos
>
>
< 1
Rn D
^
c5
>
LD (z1D)/h
>
>
(z1D)/h
>
e
J
e
B
: 2
1 R
n
if
z . 2D
if
z# 2D
.
2
^ and (›c/›z)/N
^
at z 5 2D yields a tranMatching c
scendental equation for the Rn:
Nm
L
L
L
J0 D 2 tan m J1 D 5 0.
ND
Rn
Rn
Rn
(A6)
We solve (A6) numerically, using Newton’s method.
Thus, the total streamfunction in the mixed layer
case is
SEPTEMBER 2015
LACASCE AND WANG
FIG. 8. Correlations between the predicted and modeled (left) density and (right) vorticity for (a) region 1,
(b) region 2, and (c) region 3. Shown are the results using the single exponential (black solid curves) and mixed layer
(red solid curves) solutions, as well as the correlations from the SQG portions alone (dashed curves).
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8
Lm kz
Lm kz
Lm z
b^
>
>
>
A
sinh
cosh
cos
1
1
g
z . 2D
> 1
1
<
Nm k
D
D
Rn D
.
c5
>
>
Lm J1 [LD e(z1D)/h /Rn ]
>
(
z1D
)/
h
(
z1D
)/
h
(
z1D
)/
h
>
z# 2D
I1 [LD ke
] 1 g1 e
cos
: A2 e
J1 (LD /Rn )
Rn
The constant g 1 is determined again from the surface
elevation. As this only involves the mixed layer solution,
the result is particularly simple:
g
g1 5 h
^ 2 A1 .
(A8)
f0
The solution obtained by matching ›c/›z instead at
z 5 2D is very similar. Having a continuous buoyancy at
the interface with a discontinuous N is equivalent to
having a delta-function PV sheet at the mixed layer base.
Such a discontinuity can support buoyancy anomalies
(Bretherton 1966; Plougonven and Vanneste 2010; Smith
and Bernard 2013), which in turn can permit Eady-type
instabilities in the mixed layer (Pedlosky 1987). For the
present solutions, however, the density anomaly is assumed
confined to the upper surface, precluding instability.
The constants for the SQG solution are instead
b^ ND
sinh(Lm k)I0 (LD k) 1 cosh(Lm k)I1 (LD k) ,
A1 5
D Nm
(A9)
^5
c
(A11)
The transcendental equation for the baroclinic modes is
also altered slightly, to
J0
LD
N
L
L
1 m tan m J1 D 5 0.
Rn
ND
Rn
Rn
ND e2(z1D)/hm
if
z . 2D
ND e(z1D)/h
if
z# 2D
.
In fact, this provides the most satisfactory fit to the density
profiles in the three cases considered, and the stratification
is continuous, so that density alone can be matched at
z 5 2D. But the results were as unsatisfactory, as with the
first solution described above. The density variations increased below the surface to unrealistically large values
near the mixed layer base, and the vorticity variations were
overly large at depth, so we chose to focus on the second
solution given above.
(A12)
The form is slightly different than before, and in the
limit of weak surface stratification the second term is
very small. So, in fact, the solutions are close to the zeros
of J0. As such, one can approximate R1 by LD/2.4048 in
the full solution.
Thus, the total streamfunction can be written as before, but with the approximate value of R1:
z . 2D
>
>
2:4048Lm J1 [2:4048e(z1D)/h ]
>
>
: A2 e(z1D)/h I1 [LD ke(z1D)/h ] 1 g1 e(z1D)/h cos
J1 (2:4048)
LD
(
N5
b^
, and
(A10)
D
D 5 ND k cosh(Lm k)I0 (LD k) 1 Nm k sinh(Lm k)I1 (LD k) .
A2 5
8
Lm kz
2:4048Lm z
b^
>
>
>
A
cosh(L
k)
1
cos
1
g
sinh
> 1
m
1
<
Nm k
D
LD D
The constant g1 is again determined by (A8).
A third solution was also tested, involving a double
exponential stratification profile:
(A7)
.
z# 2D
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